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Diffusive shock acceleration repeatedly returns particles to a shock wave by scattering them in the upstream and downstream plasma. For fluid speeds $u_1>u_2$, particle speed $v\gg u_1$, and shock compression ratio $r=u_1/u_2$, the cycle gain is $⟨\Delta p⟩/p=4(u_1-u_2)/(3v)$ and the cycle escape probability is $4u_2/v$. Their ratio gives $f(p)\propto p^{-3r/(r-1)}$ for the isotropic phase-space one-particle distribution function. For $r=4$ this is $p^{-4}$, corresponding to $dN/dE\propto E^{-2}$ in the ultrarelativistic regime.

Magnetic turbulence consists of irregular, fluctuating magnetic fields coupled to moving plasma. Scattering from these fluctuations can randomize a charged particle's direction in the local fluid frame and permit repeated shock wave crossings in diffusive shock acceleration.

A shock wave is a thin, irreversible compression layer across which fluid density, pressure, temperature and velocity change rapidly, while conserved fluxes satisfy the Rankine-Hugoniot conditions. In the shock frame, a compressive hydrodynamic shock converts ordered upstream kinetic energy into downstream thermal energy and increases entropy. The upstream normal flow is supersonic; the downstream normal flow is subsonic for an ordinary gas-dynamic compression shock.

In an accreting binary star, a supersonic stellar wind can be captured by a compact companion and shocked. An accretion stream striking an accretion disk, an accretion-disk boundary layer, and a magnetically guided accretion column provide other sites where infall energy is dissipated and high-energy radiation is generated. Shocks heat and decelerate material; a disk also needs transport of angular momentum before its material can reach the compact star.

In diffusive shock acceleration, charged particles scatter from magnetic irregularities on both sides of a shock and repeatedly cross the converging flows. The average energy gain is first order in the velocity jump divided by the particle speed. Escape competes with acceleration, producing a nonthermal distribution rather than a single downstream temperature. Supernova remnants are important candidate sources of Galactic cosmic rays; shocks associated with active galactic nuclei, their jets and lobes, and gamma-ray bursts are candidate extragalactic accelerators. Ultrarelativistic energies require sufficient size, magnetic field and lifetime, and losses must not outpace acceleration. In particular a relativistic particle's Larmor radius must fit within the accelerator; the existence of a strong shock alone does not guarantee unlimited particle energy.

The Doppler factors for the two discrete blobs are ${\mathcal{D}}_{\pm }=[\gamma (1\mp \beta \cos \theta ){]}^{-1}$. Their ratio is $(1+x)/(1-x)=77/33=7/3$. Lorentz-invariant specific intensity means $I_{\nu }/{\nu }^3$ is unchanged; with $\nu =\mathcal{D}{\nu }^{\prime }$ and the same rest-frame power-law spectrum,The brightness ratio at the common observed frequency therefore givesThe stated identical intrinsic specific intensities must refer to comparable intrinsic states. If the blobs fade with their proper ages, equal observer times sample different ages and need that correction. A steady continuous jet has a different boosting exponent, $2+\alpha$; the discrete ejecta and the specified specific intensity transformation justify $3+\alpha$ here.

For optically thin synchrotron radiation from an isotropic electron distribution $dN/dE\propto E^{-p}$, the characteristic emitted frequency scales as $B{E}^2$ and the total single-particle power as $B^2{E}^2$. Thus $j_{\nu }\propto E^{-p}{E}^2/(d\nu /dE)\propto {\nu }^{-(p-1)/2}$; integration over the full single-particle spectrum preserves this exponent. HenceThe test-particle strong nonrelativistic diffusive shock acceleration prediction is $p=(r+2)/(r-1)=2$ for compression ratio $r=4$, giving $\alpha =0.5$. The inferred spectrum is steeper. If interpreted literally within that uncooled formula, $p=2.50$ would correspond to $r\simeq 3$, but cooling, escape, and other acceleration conditions can also steepen it.

The observed Galactic cosmic ray differential energy spectrum below the knee is approximately $E^{-2.7}$, steeper still; its formal synchrotron radiation equivalent would be $\alpha \simeq 0.85$. Thus $p\simeq 2.50$ lies between the simplest source prediction and the propagated cosmic ray slope. The comparison is qualitative: most measured cosmic rays are nuclei, whereas the radio slope here probes electrons, and propagation and radiative losses alter the spectra differently.

Diffusive shock acceleration is viable in principle at $511\mathrm{GeV}$: repeated scattering off magnetic irregularities on opposite sides of a strong nonrelativistic shock wave can reach such energies. Actual viability requires a sufficiently large and long-lived accelerator with an acceleration time shorter than its escape and loss times; $\gamma$ alone does not establish those conditions.

Take positive velocity toward the upstream material, and let the front move at $+v_s$ in the upstream rest frame. For a strong adiabatic shock wave in a gas with adiabatic index $5/3$, the Rankine-Hugoniot conditions give ${\rho }_d/{\rho }_u=4$ and the shock-frame inflow speeds satisfy $u_1=v_s$, $u_2=v_s/4$. The signed velocities in the three frames areThus the two scattering media converge at $V=u_1-u_2=3v_s/4$. For a crossing particle with incident energy $E$ in its local plasma rest frame, a Lorentz transformation to the approaching other plasma giveswhere $\mu >0$ is the incident direction cosine measured toward the approaching medium and $pc\simeq E$. For an almost isotropic crossing population, flux weighting gives $⟨\mu ⟩={\int }_0^1{\mu }^{2}d\mu /{\int }_0^1\mu d\mu =2/3$. Scattering randomizes directions without changing energy in that plasma frame, and the reverse crossing has the same mean gain. ThereforeThese are first-order mean gains. They are not an increase of energy in one fixed frame caused by an instantaneous boundary crossing. Literally choosing the electron's instantaneous rest frame gives incident $E=m_e{c}^2$, and the other gas approaches at nearly $c$, so the expansion in small $V/c$ is invalid. This concrete frame check explains why the printed description of $V$ must instead refer to the upstream/downstream scattering frames.

Use the shock frame, with upstream and downstream fluid velocities $u_1=u$ and $u_2=u/4$ pointing in the same direction. These are the Strong-shock Rankine-Hugoniot conditions for a nonrelativistic monatomic gas with adiabatic index $5/3$. The two scattering media approach one another at relative speed $U=u_1-u_2=3u/4$, to first order in $u/c$. An energetic electron repeatedly crosses the shock wave, scatters from magnetic turbulence carried by the local plasma, and returns. Each scattering approximately conserves its energy in the local fluid rest frame while changing its direction; comparing successive fluid frames produces a systematic energy and momentum gain. The energy comes from the converging bulk flow, through the moving magnetic field irregularities and their associated electric fields. This is diffusive shock acceleration, a first-order Fermi acceleration process.

Let $\eta$ be the positive cosine of the crossing angle with the shock wave's normal. The Lorentz transformation between fluid frames gives, for either crossing direction with the corresponding sign of the normal reversed,For an almost isotropic population, the number crossing a unit area is weighted by its normal velocity. Thus the normalized crossing distribution is $2\eta d\eta$, rather than a uniform distribution in $\eta$, andFor an ultrarelativistic electron, $E\simeq pc$ and its speed $v\simeq c$, soMore generally $dE/dp=v$ gives $⟨\Delta p⟩/p=2U/(3v)$, while a nonrelativistic kinetic energy has $⟨\Delta K⟩/K=4U/(3v)$. A complete upstream–downstream–upstream cycle contains two crossings and has ultrarelativistic fractional energy gain $u/c$.

The initially cold electrons need an injection stage: shock wave heating and small-scale plasma processes must first produce particles with speed much larger than $u$ that can return across the front. The $v\simeq c$ result is a description of that accelerated population; a cold electron cannot simply be assigned $E\simeq pc$ at its first encounter.